ieee transactions on application of modern synthesis to aircraft control: three...

IEEE TRANSACTIONS ON AUTOMATIC COKTROL. VOL. AC-3 I . NO. 1 I . NOVEMBER 1986 995

Application of Modern Synthesis to Aircraft Control: Three Case Studies

DAGFINN GANGSAAS, KEVIN R. BRUCE, JAMES D. BLIGHT, AND UY-LO1 LY

Abstract-The role of feedback control in the solution of aircraft stability and control problems is discussed. It is argued that this role is becoming more and more important and is a key to meeting performance objectives for new aircraft. .4s a consequence, the control engineer must develop control laws for applications with many, sometimes conflicting, control objectives and stringent safety requirements. In the past, predominantly classical sy-nthesis techniques have been used in industry to develop control laws for aircraft. However, the so-called modern synthesis techniques that are claimed to improve quality and reduce development cost are having increased practical use in industry.

Modern synthesis techniques that offer significant promise of practical applications are discussed briefly, and three case studies of their application to aircraft control problems are presented. The first example involves the redesign of an autopilot control law to improve stability and reduce sensitivity to plant parameter variations. A much improved control law was developed, flight tested, and implemented in the autopilot of the Boeing 767 commercial transport airplane. The second and third examples address the development of control laws for aircraft that rely extensively on feedback control to furnish satisfactory stability and control characteristics. These two applications are typical of the next generation of transport aircraft that will rely extensively on feedback control to improve fuel efficiency. The control laws gave the airplane flight characteristics that are superior to those of current airplanes.

The solutions presented could have been obtained using classical synthesis techniques. However, the modern approach reduced the number of design iterations required and appeared to produce better control laws for a given level of practical experience of the control engineer. In our opinion, this approach to control law synthesis will play an increasingly important role in control design for present and future aircraft. Implemented in a user-friendly engineering workstation environment, these techniques offer improvement in quality and reduction in development cost, and for some applications, particularly to future high- performance aircraft. only the modern multiloop synthesis techniques will offer practical and cost-effective solutions.

INTRODUCTION

T HIS paper highlights the importance of the role of control law synthesis in the design of new aircraft and demonstrates

through three examples how modern computer-aided synthesis techniques offer reduced development cost and improved quality of the control laws. Our thesis is that when these techniques are combined with the good understanding of th: control problem that has always been required for success. significant benefits accrue.

Stability and control has beer, one of the major technica! challenges facing aircraft designers. The failure of many aircraft projects in the past can be directly attributed to inadequate solutions to the stability and control problem. The success of the Wright brothers in conducting the world’s first powered flight was due to their good understanding of the aircraft control problem. As expressed in [l]. “The practical achievement of satisfactory

recommended by Associate Editor. U’. F. Powers. Manuscript received November 11, 1985: revised July 5 , 1986. Paper

The authors are with the Boeing Company. Seaitle. W-A 98124-2207. IEEE Log Number 8610659.

stability and control is probably the greatest contribution of the Wright brothers in the development of the airplane.” Today, the importance of satisfactory aircraft stability and control characteristics. and the need to incorporate them into the aircraft design is well recognized. The aircraft designer must ensure that a comprehensive set of aircraft stability and control requirements, such as those in [ 2 ] . are satisfied. Initially. such requirements were met by ensuring that the aircraft exhibited inherent or natural stability and control characteristics. This implied that the aircraft could be safely piloted using relatively simple mechanical connections between the flight control surfaces and the pilot’s controllers.

With introduction of the jet engine, aircraft speed and altitude envelopes increased dramatically. Designers found it increasingly difficult to meet the requirements. In particular, the forces required to maneuver the aircraft became excessive and beyond the pilot‘s ability. This led to the introduction of powered controls. Virtually all high-performance aircraft today rely on some form of hydraulic power actuation of the control surfaces. The pilot supplies the actuation forces indirectly by controlling the amount of power applied to the surfaces.

Having satisfactorily solved the problem of furnishing the required control forces. the aircraft also had to be designed to meet stability requirements. Furnishing satisfactory inherent stability characteristics over the full flight envelope imposed significant range and payload penalties, and for aircraft with a very wide speed envelope, such as vertical takeoff and landing, supersonic. and hypersonic aircraft, it has been impossible.

The Role of Feedback Control

The importance of feedback control in aircraft design was firmly established following World War I1 [ 11. As a consequence, during the last four decades, feedback control has become more and more a part of the solution to the aircraft stability and control problem. All high-performance aircraft produced today employ some form of feedback control to alter their stability and control characteristics. Feedback control will play an increasingly more important role in meeting the performance objectives of new aircraft. Control law synthesis and control performance analyses are becoming part of the iterative aircraft design cycle and influence the airframe and propulsion system configuration to ensure the best possible performance benefits. This is in contrast to past practices where the control design would mainly accom- modate given airframe and propulsion system characteristics.

The feedback control functions of the modem fighter aircraft and the Space Shuttle, for example. are necessary for safe flight. This will also be true for most aircraft in the future, including new transport aircraft. The control laws must function properly at all flight conditions and aircraft states. Thus. they must not only provide nominal performance, but also ensure safety of flight.

Solving the Control P,.obletn

Control problems combining demanding performance objectives and stringent safety requirements must be addressed. In the past these problems have been solved predominantly using

0018-9286/86/1100-0995$01.00 @ 1986 IEEE

996 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-3 1, NO. 1 I . NOVE-MBER 1986

classical single-loop frequency response and root locus design techniques. This approach to aircraft control design is essentially the same as that outlined in [ 11 in 1951. Although the methods have been successful and are generally accepted, particularly in industry, researchers nonetheless have devoted considerable effort to the development of new so-called modern synthesis and analysis techniques. However, for more than 15 years there has been debate among researchers and practicing control engineers in regard to the most appropriate new approach.

The various proposed methods can be divided into two schools of thought. One is based on frequency domain descriptions of the physical plant. control objectives, and sensitivity properties. It is derived from the classical single-loop design methods of Bode, Evans, and Horowitz [3]-[5] that have been extended to multiloop problems (notably 161 and 171). The second is based on time- domain state-space descriptions of the physical plant. The corresponding control objectives are expressed as time-domain response criteria. The most popular synthesis approach is based on linear quadratic optimization. which has a particularly well- developed theoretical foundation [SI-[ 131. There are precise mathematical relationships between the frequency-domain and timedomain approaches [ 141. However, there has been a ten- dency among engineers to teach and practice either one or the other.

Multiloop frequency domain techniques are claimed to provide a natural framework for implementing practical design requirements. They would presumably draw upon the extensive frequency domain expertise and insights derived from single-loop systems. However, the techniques have failed to find widespread use among classical control engineers in industry. The latter still rely predominantly on one-loop-at-a-time frequency response and root locus techniques.

The proponents of modern time-domain syntheses claim these techniques can handle multiloop control problems in a formal and systematic manner. Many papers dealing with application of modern control techniques have been published in recent years. Unfortunately, many of these are only of academic interest as they are highly theoretical and lack focus on practical design. This has made it difficult for the control engineers to put new theory into practice. Thus. in spite of the availability of good computational software. the techniques have not found widespread practical use in industry. This failure is. for the most part. due to 1) preoccupation with mathematical rigor and the notion of time domain optimality. 2 ) insufficient understanding of the relation- ship between design requirements and the mathematical formulation of the solution. 3) failure to recognize the inherent control performance limitations imposed by the nature of the physical plant. and 4) lack of attention to the effects of uncertainty in the plant model description.

The Approach

These shortcomings have been recognized over the last ten years. In particular, the problem of plant model uncertainty has received considerable attention and motivated significant applications-oriented research. It has led to developments that offer multiloop synthesis and analysis with much the same ease and reliability as the classical techniques for single control loops. This can largely be credited to the successful bridging of the gap between theory and practice. At the risk of overlooking other good approaches. only two will be highlighted here.

The first approach probably represents the most significant advancement in the development of practical modern synthesis and analysis techniques in recent years. It is based on (1 ) the extensions of single-loop frequency domain shaping techniques to multiloop systems [15], and ( 2 ) the adaptation of the linear quadratic Gaussian (LQG) method for the synthesis of the required frequency domain shapes of the multiple control loops [ 161 and [ 171. The procedure takes advantage of the well-known robustness properties of the linear quadratic regulator [ 181 and its

dual, the linear time-invariant Kalman filter [ 191. It uses the loop recovery procedures [20] and [ 1 I ] to obtain the desired frequency domain loop shapes at the plant inputs and outputs. respectively, for the combined plant and LQG controller. In essence. this approach has generalized for multiloop systems the following four fundamental principles of classical single-loop synthesis: 1) high loop-gains within the control bandwidth for control performance, 2 ) well-behaved crossovers for good stability properties. 3) low loop-gains outside the control bandwidth for insensitivity to modeling errors. and 4) good understanding of the fundamental limitations imposed by nonminimum phase and lightly damped plant zeros.

A second approach based on parameter optimization. but of a more general nature than the standard LQG procedure. will be addressed in one of the case studies in this paper. This method, which is described in [?I]. has the following features: 1) direct synthesis of low-order controllers of arbitrary structure. 2 ) direct synthesis of a fixed or gain scheduled controller for multiple plant conditions. and 3) incorporation of design requirements via linear and nonlinear equality and inequality constraints. In addition, it can be used in conjunction with the frequency domain shaping procedures referenced earlier. Computationally. it is a more cumbersome and costly procedure than the LQG-based approach, but it can be used with success for many classes of control problems.

CASE STUDIES

Three case studies of the application of modern control law synthesis to aircraft control are presented. All three involve one control input and multiple sensors. These are typical of aircraft control problems solved in the past using classical synthesis. They do not represent applications of modem synthesis techniques at their point of strength. which is solving problems with many highly coupled control loops. However. the case studies will demonstrate that the new techniques offer significant benefits even in the case of single input systems. The work should be viewed as typical for industry engineers who are learning the new techniques by applying them to traditional control problems. In our opinion, this is a necessary and important experience for engineers in industry prior to addressing the more complex multiloop problems.

A list of nomenclature is given in Appendix A . For those readers who are not familiar with aircraft. a brief description of terms associated with flight mechanics and control is given in Appendix E. State models used in Examples I1 and 111 are given in Appendix C. The state models used in Example I involved over 50 states due to very detailed modeling of the aircraft, sensors. computers. control laws. servos. and actuators. It is outside the scope of this paper to include all this model data.

Case Study I: Improvement of the Boeing 767 Lateral Autopilot

This example addresses the elimination of a small amplitude limit cycle instability experienced on the Boeing 767 commercial transport airplane. The problem was associated with the heading and track hold autopilot. called the lateral autopilot, and was not solved after repeated attempts using classical synthesis techniques. The solution involved a good understanding of the control problem combined mith a straightfonvard application of linear quadratic regulator theory. The latter furnished the necessary insight. in terms of required feedback signals and corresponding gains. to eliminate the limit cycle instability without compromising the performance of the autopilot heading and track hold functions. The success can be attributed to the excellent robustness properties of the regulator [18]. The data presented here are summarized from an earlier paper 1221.

Problem Statement: Occasional ride discomfort was reported during early passenger service of the Boeing 767 commercial jet

GANGSAAS et al.: MODERN SYNTHESIS AND AIRCRAFT CONTROL 997

transport. It was due to a small-amplitude, sustained yawing oscillation that occurred only during high altitude cruise flight when both the yaw damper and lateral autopilot were engaged. The yaw damper increases the damping of the dutch roll mode of the aircraft using the rudder as a single control. The dutch roll mode, which involves yaw and roll angle oscillations, is described in Appendix B. The yaw damper is normally engaged both in manual and automatic flight. During automatic flight, the lateral autopilot is engaged and it controls heading or track angle using the combination of left and right ailerons as a single control.

Flight testing showed that if the yaw damper was disengaged, that is, the rudder control loop was opened, but the lateral autopilot was engaged, the aircraft did not exhibit the limit cycle instability. However, with this nonstandard configuration of the yaw damper and lateral autopilot, the aircraft dutch roll mode was lightly damped. Analysis of flight test data showed that engaging the lateral autopilot tended to reduce the damping of the dutch roll mode particularly with the yaw damper disengaged. Fig. 1 shows the open-loop gain and phase characteristics in the aileron control loop with the rudder loop open. It is clear that the stability margins are very small and that very small gain and phase variations would lead to instabilities at the dutch roll mode frequency. Based on this, it was hypothesized that dead band and hysteresis in the rudder control loop combined with relatively small variations in aerodynamic control effectiveness in the aileron loop could cause the observed limit cycle oscillations.

Tests had shown that all of the aerodynamic parameters and nonlinearities were well within normal and predicted values for those aircraft exhibiting the limit cycle behavior. Thus, it appeared that due to adverse coupling between the aileron and rudder control loops, there was high sensitivity to small nonlinearities and variations in aerodynamic parameters. It was further hypothesized that this sensitivity could be reduced if the destabilizing effect on the dutch roll mode from the lateral autopilot was eliminated, or even better, turned into a stabilizing effect. Thus, the design problem was to improve the dutch roll damping and improve stability margins with the lateral autopilot engaged and the yaw damper disengaged.

The autopilot control law had been synthesized using standard root locus and frequency response techniques with sequential loop closures on the various feedback sensors. The latter comprised almost the full state vector except for sideslip angle 0, and yaw rate r. Yaw rate was sensed. but not used for feedback. Root locus analysis showed that dutch roll damping could be improved. However, this was at the expense of reduced heading or track mode stability that led to significant degradation in lateral autopilot performance. Extensive root locus analysis failed to produce a set of gains that offered significant improvements in dutch roll mode stability while maintaining the required lateral autopilot performance. It was then decided to use full-state feedback synthesis in an attempt to establish whether or not a better solution existed.

Objectives and Constraints: The objective was to eliminate perceptible residual yaw

oscillations, without affecting lateral autopilot performance. Reduce rms lateral accelerations and aileron deflections due

to gust inputs. Due to cost and schedule constraints only changes to the

control laws in the lateral autopilot were allowed. The lateral autopilot had to operate satisfactorily with and

without the yaw damper engaged. Controlling either heading angle or track angle should not

require gain changes. The control performance and stability had to be insensitive to

nonlinearities and variations in the aerodynamic characteristics. The control loop bandwidth and high-frequency gain should

not be greater than that of the existing design. Design Method: Fig. 2 shows a block diagram of the plant

model used for analysis and synthesis. It represents the aileron actuation system. flight control computer time delay, airplane

0 ; 1 ' \ 10 100 FREOUENCY. Irad:rl

\ ,,&DUTCH ROLL MODE FREOUENCY

_u

4

Fig. 1

0 J -40

-80

-120

160

200

-240

-320

360 01 1 1 10 100

FREOUENCY. (radlrl

. Aileron open-loop frequency response with rudder

C O M M A N D

ANTI -AL IASING F ILTERS

loop open.

SENSORS

I : .

R U D D E R D E F L E C T I O N

Fig. 2. Plant model-Case Study I .

dynamics, and antialiasing filters on the sensor signals. In addition, the yaw damper control laws. computational delay, sensors, and rudder servo and actuator dynamics were modeled. This total model was expressed in state-space form at various flight conditions (Fig. 3). The state vector comprised over 50 elements. The rudder control loop was open for control law synthesis (see Fig. 2 ) , however, it was closed for performance analysis with the yaw damper engaged.

The control law design was based on the airplane model for the nominal cruise flight condition, using linear quadratic regulator

998 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-31. NO. 11. NOVEMBER 1986

(ALTITUDE = 40,000 f t )

Flight ( Ib) conditions Weight,

Center of gravity, Mach

(% MAC1 no.

1

2

0.82 36 180,000

0.86 36 300,000 4

0.75 36 300,000 3

0.73 13 300,000

Fig. 3 . Flight conditions-Case Study I.

(LQR) synthesis. Full-state feedback gains were calculated based on the following cost function:

~ = ( l / 2 ) ~ [ Q r ( ~ ~ - ~ ) ~ + Q $ ( ~ ~ - ~ ) 2 + ~ ~ 6 ( ! ( ~ ~ - ~ ) ) ~

+ Q d r ( ~ d r ) ' + ( L ) ' l (1)

where Qr, Qd, Q,$, and Qdr are the penalty weighting on yaw rate error, complemented heading or track error. integral of complemented heading or track error, and dutch roll mode displacement ydr, respectively. For this problem. measured yaw rate r is approximately equal to heading rate $ [23]. The dutch roll mode displacement ydr is related to the states through the eigenvectors in a standard modal decomposition. 6, is the input to the aileron actuators. The subscript c in ( 1 ) refers to command values.

To meet the requirement that heading angle $ or track angle $,r

should be controlled interchangeably without control law gain changes. it was necessary to close the proportional and integral loops on complemented heading or track angle $. as defined in Fig. 4. Angles $ and $fr are related by:

$ f r = $ + B . (2)

The dutch roll mode dominates the slideslip f l response. Thus. if $fr were substituted directly for $ there would be a significant impact on the dutch roll mode stability requiring control law gain changes from a redesign. By setting the break frequency (a = 0.2 rad/s) of the complementary filter in Fig. 4 well below the dutch roll mode frequency of 1 rad/sl there is sufficient attenuation of the p response at this frequency to ensure minimal impact on dutch roll mode stability when is substituted for $. The yaw rate r input to the complementary filter ensures that good heading and track mode stability is maintained. This is a good example of how frequency domain loop shaping can be used to help satisfy apparently conflicting design requirements.

Reflecting a standard rate, proportional. and integral control structure, the penalties Qr , Qi, and Q, were adjusted to obtain the same heading or track mode damping, bandwidth. and integral time constant as the classical design. Next the damping of the dutch roll was increased by increasing the dutch roll mode weighting Qdr. This did not affect the damping of the other modes. For example, the heading or track mode poles remained unchanged as the dutch roll damping changed from 0.065 to 0.17 (at 1.01 rad/s).

The gain on sideslip angle /3 changed from a positive value to a large negative value as Q, was increased. Unfortunately$ f l was not available as a feedback sensor. Qdr was therefore adjusted to give a set of gains that included a gain of zero on 8. This resulted in a dutch roll damping of ldr = 0.08. Although the absence of sideslip feedback limited the amount of damping that could be obtained. the improvement was significant when compared to the dutch roll damping of Cdr = 0.01 for the classical design.

Fig. 5 compares the gains obtained from the LQR synthesis with those of the classical design. Only the significant gains from the full-state LQR design were retained. The remainder were set

H E A D I N G A N G L E . C, OR COMPLEMENTED HEADING. TRACK ANGLE, +fr OR TRACK ANGLE. 6

+

--+p YAW RATE, r

Fig. 4. Complemented heading or track angle

Control laws

Original 7 1 4

Fig. 5. Gains used in flight test.

to zero without any impact on stability and control performance. The proportional heading and integral heading gains were approximately the same for both designs. However, there are significant differences in the two designs for the yaw rate r, roll angle 4. and roll rate p , gains. The classical design had a zero gain on yaw rate while the LQR design has a relatively high gain. This gain maintained a fixed ratio to the heading gain for all designs having a well-damped heading mode. The roll angle gain was reduced by a factor of three and the roll rate gain was reduced by 30 percent.

There were no combinations of weights in the cost function (1) that would produce a roll angle gain as large in magnitude as that of the classical design. Originally. the rationale for this large gain was to ensure good tracking performance for heading angle. In coordinated flight. that is. with the sideslip angle close to zero, roll angle and yaw rate are related kinematically [23]. and therefore are equivalent feedback signals. However, this is not true when there are significant sideslip oscillations as in the case of a lightly damped dutch roll mode. It is interesting to note that the LQR synthesis provided the insight that yaw rate feedback rather than roll angle feedback would give a much better tradeoff between robustness and control performance as will be seen later. The fact that yaw rate feedback had been excluded in the earlier work using the root locus technique accounts for the failure to find an acceptable solution.

Performance: Prior to flight test the control law performance was evaluated by analysis at ten different flight conditions. These reflected the full range of gross weight. center of gravity location, and speed expected in high altitude cruise flight. Data from the four worst flight conditions are presented here. Analysis was performed with the yaw damper both engaged and disengaged and with control of heading angle or track angle. All combinations produced satisfactory results [22]. However, only data for heading control with the yaw damper disengaged will be presented here since it represented the most difficult design problem.

Fig. 6 shows the damping of the dutch roll and heading modes. The redesigned control law shows a significant reduction in the sensitivity to variations in flight condition. The original control law has minimum damping of Cdr = 0.01 for the dutch roll mode and = 0.47 for the heading mode. Over the same range of flight conditions the new control law furnished minimum damping of cdr = 0.08 and [,, = 0.7 for the corresponding modes. In fact. for all cruise flight conditions the modes exhibited excellent stability without any gain scheduling [ E ] .

One of the aerodynamic parameters that introduces coupling between the rudder and aileron control loops is the yawing moment derivative with respect to aileron deflection Cnba. Its value is difficult to predict and may vary considerably including reversing the sign at some flight conditions. Thus. one of the design requirements was that the dutch roll damping should be insensitive to changes in this parameter. Fig. 7 shows the variation

GANGSAAS et ai.: MODERN SYYTHESIS AXD AIRCRAFT CONTROL

0 CLASSICAL CONTROL LAW

MODERN COUTROL LAN

1 2 3 1 1 2 3 4

FLIGHTCONDITIONS FLIGHTCONOITIONS

Fig. 6. Mode damping.

DUTCH Y A W D A M P E R D I S E N G A G E D D A M P I N G

\ (Cdrl / F L I G H T C O N D I T I O N

M O D E R N C O N T R O L L A W 3% CLASSICAL CONTROL LAW

1 0 1 1

Fig. 7. Sensitivity in d u t c h roll mode damping.

of the dutch roll damping to large changes in Cnao. The original control law exhibits considerable sensitivity to these changes while the new control law shows significantly less variation in the dutch roll mode damping.

Another design objective was to minimize gust response in particular lateral accelerations in the cabin and commanded aileron deflections. Fig. 8 shows rms lateral accelerations at three positions in the cabin. and rms aileron deflection. bank angle. and heading angle. The new control law offers significant reductions in all rms responses. Of particular importance is the 60 percent reduction in rms lateral acceleration in the aft cabin and the 69 percent reduction in rms aileron deflections.

Flight Test Results: The new lateral autopilot control law was implemented in the flight control computers. This entailed modifying the autopilot gain schedules to equal the redesigned gain values of Fig. 5 at the cruise flight conditions and adding a yaw rate feedback to the aileron command input. The performance of the original and new control laws were evaluated during flight test. Fig. 9 shows the light dutch roll damping with the original control law. Fig. 10 demonstrates the significant improvement in dutch roll damping offered by the new control law. This improvement was demonstrated over a wide range of flight conditions with and without the yaw damper engaged. The particular test aircraft had never exhibited the limit cycle behavior with the original control law. However. it was conjectured that this demonstrated improved dutch roll mode damping would eliminate the problem from those airplanes exhibiting limit cycle oscillations in service.

The new flight control law was incorporated on airplanes that earlier had exhibited the limit cycle oscillations. Pilots who flew with the modified autopilot control law gave favorable comments and said that they now did not detect any limit cycle oscillations. They considered performance of the autopilot with the new gains

-

2 -

1 -

0 -

to be a

6,

yific ar

1 5 f m R M S L A T E R A L G U S T

YAW DAMPER D ISENGAGED

L O C A T I O N

Fig. 8. rms turbulence response.

A F T

It improvement. The modified control law is now incorporated as a permanent change to the autopilot.

~~ ~

Case Study 11: Control Law for Longitudinal Control of a Modern Transport Airplane

This application involves the synthesis of a command and stability augmentation control law for a transport airplane with relaxed requirements for inherent longitudinal stability. This airplane is typical of the next generation of transports. The design involved application of frequency-shaped LQG synthesis. The control law performed well during nonlinear piloted simulations. It was derived from a single-point design and achieved good control performance and robustness properties over the full flight envelope and center of gravity range using minimal gain scheduling. The work has been summarized previously [24].

Problem Statement: Traditionally, transport airplanes have been designed to have a certain level of inherent longitudinal stability. This and other control requirements dictate the size of the horizontal tail and restrict the permissible most aft location of the center of gravity (c.g. j . The efficiency of these airplanes can be improved by decreasing the horizontal tail size and moving the c.g. aft. The corresponding reductions in weight and trim drag from the decreased tail size and trim load on the tail can yield a significant reduction in fuel consumption [25]. However. these airplanes will have unsatisfactory longitudinal stability and control characteristics within part of their c.g. and flight envelopes.

The stability and response characteristics for such an airplane were evaluated at the four flight conditions listed in Fig. 11. Fig. 12 shows for a range of c.g. locations typical normal acceleration and pitch rate time responses to a step elevator input. Fig. 13 shows the corresponding long-term speed responses. The responses are typical of all four flight conditions. In Fig. 14 the corresponding eigenvalues are listed. Except for the landing flight condition, the airplane is unstable with the c.g. at the aft location. A command and stability augmentation control law is required to provide satisfactory airplane stability and control characteristics.

Requirements and Objectives: The following are the main design requirements and objectives.

Satisfying flying qualities requirements [2] and [26]. These are detailed specifications for allowable stability and output response characteristics.

Furnish constant or task-tailored pilot column force gradients with respect to commanded normal acceleration and airspeed changes across the c.g. range and flight envelope. These are

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-31, NO. 11, NOVEMBER 1986

Flight Ten With Yaw Damper Off

TIME, ( 5 ) TIME. I s 1

Fig. 9. Flight test results with classical control lam,.

TIME, (11 TIME, Is1

Fig. 10. Flight t a t results with modem control law.

VFCXFC

25.5C-l

deg "I1

184,000 lb

Fig. 11. Flight conditions-Case Studies I1 and III.

VMlN F L I G H T C O N D I T I O N

TIME, ( 4

- --- C.G. LOCATION:

50% MAC 18% MAC

Fig. 13. Speed response of open-loop airplane. VMlN FLIGHTCONDITION

0 2 4 6 8 I 10

TIME. 1s) C.G. LOCATION: -- - - 1 8 8 MAC - 50% hlAC

Fig. 12. Normal acceleration and pitch rate response of open-loop airplane Fig. 14. Eigenvalues of the opedoop airplane-Case Study 11.

GANGSAAS er al.: MODERN SYNTHESIS AND AIRCRAFT COKTROL 1001

important parameters in determining the airplane's controllability F,

[2] and [26]. COLUMN FORCE s t 1

The short-period and phugoid mode damping ratios must be '

greater than 0.5. These modes are described in Appendix B. Normalized pitch rate response to step column force input

should fall within a particular envelope (shown in Fig. 34). Turbulence and wind shear responses must be as good as or

better than current airplanes. Satisfy k 10 dB gain margin and k45 deg phase margin

within the control bandwidth. The loop gain crossover frequency must not exceed 3 rad/s

and the high-frequency loop gain must be below - 10 dB at 10 rad/s with a minimum of - 40 dB/decade slope beyond 10 rad/s to Fig. 15. Synthesis model for LQR design. avoid destabilizing unmodeled structural modes at these higher

IDEAL MODEL

-1 K F 1 "ZYC

MODEL

HIGH-PASS FILTER

- 1 $2 I VY - 1 S2+141r+10O2 I '

frequencies. Gain scheduling must be functions of easily measured

parameters. Control Law Synthesis: The airplane and wind dynamics were

described by linear time-invariant state-space models for the four flight conditions. The control law synthesis was performed based on the CRUISE flight condition with the center of gravity at the most aft location. This condition was selected because it had the most unfavorable input to output phase characteristics at the expected crossover frequency (see Fig. 37). Analysis was performed and the gain schedules were developed based on all flight conditions described in Fig. 11. The synthesis was accomplished using linear quadratic Gaussian (LQG) synthesis with loop shaping [8]-[11], [Is]-[17]. [27]-[29].

Fig. 15 shows the model used in the LQR synthesis. The airplane model includes the aircraft longitudinal dynamics and control servo and actuator dynamics. The disturbance model consists of longitudinal and vertical Dryden turbulence models [30] and a horizontal wind shear model. In addition, an ideal column force command model defining desired transient and steady-state response characteristics to pilot inputs and a model of a high-pass filtered output of the control input are included. The purpose of the latter is to allow adjustment of the control loop gain rolloff characteristics at high frequencies. The total synthesis model is given in Appendix C.

The gains were calculated to minimize the cost function:

J = (1/2)E[Q,yt+ Qc~>f+6:c]. (3)

This cost function was constructed to reflect the design requirements for 1) transient and steady-state command response characteristics. 2) good turbulence and wind shear response, 3) insensitivity to model errors and parameter variations within the control bandwidth, 4) robustness with respect to unmodeled dynamics outside the control bandwidth, 5) well-behaved crossover characteristics, and 6) good damping of all modes.

The control input was the elevator servo command tiec. There were two output criteria variables yN and y,. The criterion yu was included to adjust the high-frequency gain attenuation in the control loop. The criterion y , represents the regulated output. It comprises a combination of the errors in mean airspeed A V,, and normal acceleration Anz, with integral control added as shown in Fig. 15.

For a stable airplane in wings-level flight, a small column force input producing an elevator deflection will result in an initial incremental normal acceleration that returns to zero and a slower speed response that settles to a new steady-state value. The sensitivities between the column force input and 1) the normal acceleration response (Ib/g) with the airspeed unchanged, and 2) the long-term airspeed response (lbjknot) with the incremental normal acceleration unchanged, are key parameters in determining the flying qualities of an airplane. They must lie within certain bounds [2].

Using the control law structure defined in Fig. 15. the feedforward gain KF (g/lb) and speed feedback gain K,, (g/knot) completely define these parameters for a stable, closed-loop

system due to the presence of the feedback integrator. Based on the guidelines in [2] and piloted simulations, column force gradients of 30 Ib/g and - 114 Ib/knot were selected for this application. The corresponding values for KF and K,, were 1/30 (g/lb) and - 1/120 (glknot)? respectively. For other applications, these parameters can be changed to reflect unique flying qualities requirements for specific pilot tasks and flight phases. The remaining gains, Ki and K,,, and the location of the sensor measuring normal acceleration nZ, were adjusted to obtain good frequency domain loop shapes between the control input 6, and the regulated output y C .

Fig. 16 shows the frequency response between the elevator and the normal acceleration measured at a fonvard location. For a small perturbation longitudinal airplane model, there is a zero at the origin [31]. This implies that for small inputs, nonzero normal acceleration cannot be maintained in the steady state using the elevator. There is also a pair of lightly damped zeros at a frequency of approximately 3 radls. These zeros control the frequency and damping of the closed-loop short period mode and '

would result in poor damping of the mode as the loop gain is increased. A zero locus was calculated as a function of the longitudinal position of the normal acceleration sensor. At a location just forward of the c.g.. the zeros are located on the real axis at - 16 rad/s and 60 rad/s outside the expected control-loop bandwidth. The corresponding frequency response is shown in Fig. 17. This loop shape will ensure good closed-loop characteristics of the short period mode.

Fig. 18 shows the frequency response between the elevator input and the airspeed output. For a small perturbation longitudinal airplane model, there is significant gain at low frequency [31]. This implies that speed can be controlled in the steady state from the elevator. Combining speed feedback with normal acceleration feedback provides the required nonzero gain at zero frequency. Fig. 19 shows the frequency response between the elevator and the output nzr,. The latter is a linear combination of mean airspeed error and normal acceleration defined as:

nru = n, + K,, Kc, A V, . (4)

K, was selected together with KF (see Fig. 15) to provide the desired steady-state column' force gradients as described earlier. K,, is the conversion factor between true airspeed in fi/s and calibrated airspeed in knots.

Mean airspeed error is defined as:

A v,, = vn, - V R ( 5 )

where VR (= U,) is the reference trim airspeed and V, is the mean airspeed. The latter is defined as:

vm = u- mu. (6)

where U is the forward speed of the airplane and urn, is the mean horizontal wind speed. In contrast, the true airspeed VT is defined as:

vT= u- !.4m,u- ug (7)

1002 IEEE TRANS.ACTIOKS ON AUTOX1.ATIC CONTROL. VOL. AC-31. NO. 1 1 . NOVEMBER 1986

CRUISE FLIGHTCONDITION

En

a,

40 - E M z 4 0

20

40

Ml 001 01 1 1 1c. ! 00

F R E O U E N C Y , I r a d k l

Fig. 16. Frequency response between elevator command and normal acceleration at forward location.

CRUISE F L l G H T C O N D l T l O h

80

a,

40 - s 20 i z o

-20

40

60 001 01 1 1 17 100

F R E Q U E N C Y Irad'rl

Fig. 17. Frequency response between elevator command and normal acceleration at mid location.


80

a,

40 m - 220 ?i 9 0

20

30

-60 001 01 1 1 10 100

F R E O U E N C Y , l w l h b

Fig. 18. Frequency response between elevator command and airspeed.


80

a,

40

20

0

20

40

-a, 001 01 1 1 10 100

F R E O U E N C Y lradirl

Fig. 19. Frequency response between elevator command and tz:#,,

where u g is the zero mean horizontal random gust velocity. The reason mean airspeed rather than true airspeed is controlled is to reduce elevator activity due to horizontal gust inputs. V,,, cannot be measured directly and therefore the estimate V, was used for control law implementation.

To meet the requirement for insensitivity to model errors and

parameter variations within the control bandwidth, an integral term was added to the output criterion as follows:

I,,z,, = Ana,( 1 + Ki / s ) (8)

where An;,, is defined in Fig. 15. A value of Ki = 1.5 was selected to ensure good integral

control at frequencies at or below 1.5 rad/s. As a result of the zero placed at - 1.5 radis. the closed-loop integral pole will move asymptotically to this value as loop gain increases. The resulting loop transfer function is shoun in Fig. 20. There is a pair of lightly damped zeros at a frequency of approximately 0.06 rad/s. These zeros control the damping and.frequency of the closed-loop phugoid mode. The location of these zeros can be changed to a more stable location by adjusting the parameter K,. However, this would change the column force to airspeed gradient away from its desired value. To avoid this. a new airspeed term was added to the output criterion as follows:

y, = I,,;,, + K, A V, . (9)

For this application, K,, was adjusted to provide damping of Cph = 0.707 for the closed-loop phugoid mode. For other applications, the gains K,, and K, can be adjusted to reflect different requirements for force gradients and phugoid stability characteristics. These can be tailored to specific pilot tasks and flight phases.

Fig. 21 shows the frequency response between the elevator command and the regulated output criterion. This loop shape will furnish the desired characteristics in terms of high gain within the control bandwidth. well-behaved crossover, and good damping of the closed-loop modes. The other criterion output. yi,, will ensure additional gain attenuation as required at and beyond a frequency of 10 rad/s.

A full-state control law was synthesized based on the cost function represented by (3) with the penalty weights Qu and Qc as design parameters. These were adjusted to furnish the required elevator loop crossover frequency (between 2 and 3 rad/s) and high-frequency gain attenuation. Fig. 22 illustrates that the LQR design meets the requirements for high elevator loop gain at low, frequencies. good gain and phase margins. and the required high- frequency gain attenuation. The values for the corresponding penalty weights are given in Fig. 23. Feedback gains are shown in Fig. 24.

The advantage of using LQR synthesis rather than root locus analysis to adjust the gains on the various states is best illustrated with an example. Suppose, starting with the current design, we want to reduce the integral control time constant. Using LQR synthesis this is accomplished by adjusting the parameter K , in the regulated output y c . Fig. 25 shows that the short period damping and the control loop phase margin remained unchanged as the integral pole is moved from - 1.4 to -2.8. However, to accomplish this the LQR synthesis furnished a solution that adjusted all the gains. not only the integral gains (Fig. 26). This, of course. is necessary to maintain good stability margins. Adjusting the integral gain only using the root locus technique has an expected destabilizing effect as illustrated in the third column in Fig. 25. To recover the short period damping and the phase margins will require iterative adjustments of all gains until the same solution as that furnished by the LQR synthesis is obtained. Thus. to obtain faster integral control, the root locus technique requires iterative adjustments of many parameters while the LQR synthesis only required the adjustment of a single parameter. Comparison of the significant gains of the two LQR designs appears in Fig. 26. It is obvious from these that maintaining good stability requires a large increase in control loop bandwidth.

Having obtained the desired elevator loop shape in the LQR design. a state estimator was synthesized. The objective was to design a feedback compensator that with the available measure- ments would furnish 1) the same elevator loop shape as the LQR designs, and 2) estimates of the horizontal and vertical turbulence

GANGSAAS el al.: MODERN SYNTHESIS AND AIRCRAFT CONTROL 1003

CRUSE FLIGHTCONDITION

001 01 1 1 10 FREQUENCY. Irad:si

100

Fig. 20. Frequency response between elevator command and Inzu


FREOUEhCY, i r a d k l

Fig. 21. Frequency response between elevator command and criterion output.

CRUISE FLIGHTCONDITIOIV

Lo;* FREQUENCY GAIN BOUNDARY

20

40 - -

.cot 01 1 1 10 1M FREQUENCY, (rad!%]

-320 360 .001 3 10 100

FREQUEhCY ( r a d 4

Fig. 22. Elevator open-loop frequency response for LQR design.

Fig. 23. LQR design parameters.

I STATE

-361

1.425

.244

I ug

I ‘Vvg 1

4 2 4

..Om 4 0 0 3

,432

4 . 4 3 7

Fig. 24. LQR feedback gains-Case Study 11.

INCREASED INTEGRAL GAIN NOMINAL DESIGN ROOT LOCUS LOR DESIGN

I I INTEGRAL POLE LOCATION -1.4 -2.7 .2.8

SHORT PERIOD DAMPING 0.67 0.13 0.67

Fig. 25. Increased integral gain-LQR versus root locus method.

Fig. 26. Comparison of linear quadratic regulator gains-Increased integral penalty.

velocities. the mean airspeed, and the mean wind speed. The former would ensure good control-loop stability margins while the latter was used to furnish good airplane responses to turbulence and wind shear in terms of low rms and peak airspeed variations, rms elevator activity, and rms normal acceleration for good ride qualities.

The model- used for the statc cstimator design is shown in Fig. 27. The feedback integrator, the ideal command response model, and the high-pass filter used in the LQR design (see Fig. 15) were not included in this model. The associated states are available directly and need not be included in the state estimator. It can be easily demonstrated that as long as their contributions to the control input are accounted for in the formulation of the LQG compensator, the separation theorem [ 101 is still valid when they are combined with the state estimator.

The process noise and sensor noise spectral densities used are shown in Fig. 28. These were set as a compromise between robustness and airplane response to turbulence and wind shear. A key tradeoff in the design was the rms elevator activity in turbulence versus peak airspeed deviations in wind shear. For the ideal case of full-state feedback the combination of low horizontal gust gain and high wind shear gain allow low elevator activity in turbulence combined with small peak airspeed deviations in wind shear. The performance obtained with the full-state design shown

IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-31. NO. 1 I , NOVEMBER 1986

WHITE PROCESS NOISE SOURCE!

W" VERTICAL MODEL GUST NOISE

U" HORIZONTAL GWST NOISE

MODEL

HORIZONTAL

VELOCITY -

WINO SHEAR NOISE HORIZONTAL WIND SHEAR RATE

ELEVATOR COMMAND INPUT NOISE

AIRPLANE DYNAUICS AND ACTUATOR

SPEED EQUATION INPUT NOISE I

'ERTICAL ;UST ELOCITY

WHITE SENSOR NOISE SOURCES . I

- Fig. 27. State estimator synthesis model.

Fig. 28. Noise spectral densities for estimator design

in Fig. 29 cannot be obtained with a LQG compensator. The figure shows the tradeoff of turbulence versus wind shear performance for several state estimator designs with estimated mean airspeed error A V,,, replacing A V,, in the feedback variable nzu. For these designs all process noise intensities were held fixed at the values in Fig. 28, except that the wind shear noise spectral density S, was allowed to vary as shown.

The LQG control law does not have wind shear rate information to feed back directly to the elevator. It uses instead a wind shear estimate based on filtered sensor data. The airspeed sensor measures both mean wind and turbulence. The estimator com- bines airspeed and longitudinal acceleration to produce estimates of wind shear and turbulence velocities. Relatively small values of the spectral density S, produce a slow wind shear estimate. The result is that the airspeed is heavily filtered and elevator activity in turbulence is thus attenuated. The elevator response in wind shear is slow resulting in large speed deviations. Conversely. relatively large values of S, produce a fast wind shear estimate and good control in wind shears. However, this results in increased elevator activity in turbulence. The method employed allows a straightfor- ward tradeoff of elevator activity versus performance in wind shear. The selected design shown in Fig. 29 was based on the maximum acceptable level of elevator activity in turbulence. The

1 TRUE VT

/\IRSPEEO

LONGITUOINAL ACCELERATION

NORMAL ACCELERATION 9 PITCH RATE

, "2 SENSORS

h VERTICAL SPEED

0 22 Onz = 0259 CRUISE FLlGHTCONDlTlON

Onz = r m r NORMAL ACCELERATION I tth cm VERTICAL TURBULENCE 1 ttkrm HORIZONTAL TURBULENCE

\ I Wr? HORIZONTAL WIND SHEAR

017p

PEAK CALIBRATED AIRSPEED DEVIATION, IAVmsI, O n )

Fig. 29. rms elevator rate due to horizontal and vertical turbulence versus peak airspeed deviation in horizontal wind shear.

wind shear design trades did not affect maneuver performance. The airspeed time history (unpiloted) due to a step shear input for the closed-loop airplane is shown in Fig. 30. and illustrates that the aircraft will return to its trim speed following a wind shear input.

Elevator input noise. 6," (spectral density De<"), was used to recover the full-state feedback stability margins at the input to the elevator actuator. In this way the robustness characteristic of full- state design was recovered. Analysis at the low-speed flight condition revealed that the eigenvalues associated with the estimate of the mean wind speed were lightly damped. This was due to a significant increase in the aircraft phugoid frequency at low speeds when compared to the high speed flight condition used for control law synthesis. Additional process noise u,, with spectral density V,,, was introduced as an input to the speed equation to improve the convergence of the estimate of the mean airspeed. This produced a LQG compensator that maintained good stability at low frequency for all flight conditions.

GANGSAAS et al.: MODERN SYNTHESIS AND AIRCRAFT COKTROL 1005

J

N-

20 40 60 80 1W 120 TIME

n

TlhlE

TIME

--- 18% cg 506 CQ -

Fig. 30. Closed-loop airplane response to ramp tail wind shear

r-------------____ I FEEDFORWARD CONTROLLER I

L - _ _ _ _ - _ _ _

VT TRUE AIRSPEED ;i FLIGHTPATH ACCELERATION I 6 VERTICAL SPEED q PITCH RATE nI NORMAL ACCELERATIMI

u s patent awlled for

REDUCED-ORDER COtMPENSATOR

Fig. 31. Closed-loop system structure.

The design resulted in a tenth-order state estimator that was reduced to eighth-order by residualizing the highest frequency complex mode. This reduced-order estimator was combined with the feedback integrator and second-order control-loop rolloff filter to produce an eleventh-order feedback compensator. The structure of the resulting closed-loop system is shown in Fig. 3 1. The elevator gain was scheduled as a function of dynamic pressure and flap position (Fig. 31) and was adjusted to maintain approximately the same elevator control loop gain at the various flight conditions.

Results: Typical responses of the closed-loop airplane to column force commands are shown in Figs. 32-35. These responses compare with the responses of the unaugmented airplane in Figs. 12 and 13. There are significant improvements in the stability and response. These characteristics are relatively invariant with flight condition and c.g. location [24]. It was a requirement that the given pitch rate response to a column force input fall within envelopes such as the one shown in Fig. 36. This requirement was satisfied at all flight conditions [24].

Elevator loop gain frequency responses that are typical of all flight conditions are shown in Fig. 37 and stability margins and closed-loop eigenvalues are shown in Fig. 38. The high gain at low frequency, good loop-gain crossover characteristics, high frequency attenuation, and stability are maintained at all flight conditions for all c.g. locations [24].

The control law was implemented on a motion base flight simulator. Extensive tests were conducted by three test pilots. For all c.g. locations and flight conditions they rated the flying

0 2 4 6 8 TIME. 111

10

c.G. LOCATION: ---- 18AMAC - 50% MAC

Fig. 32. Closed-loop normal acceleration and pitch rate response at CRUISE flight condition.

0 20 40 MI 80 100 120 TIME. 111

2 % .7 5 -.9 ', >,' -. -1.1 -. -_ ---- - _ _ _ _ _ _ _ _ _ _ _ _

0 20 40 MI 80 100 120 TlhlE. ( $ 1

C.G. LOCATION: ---- 18% MAC - 50%MAC

Fig. 33. Closed-loop speed response at CRUISE flight condition

0 2 4 6 8 10 TIME. (I!

0 ,

4- -8

i g -12 yl m -10

14 2 4 6 8

TIUE. (I! 10

C.G. LOCATION-

- 50% MAC

-- -- l % M A C

Fig. 34. Closed-loop normal acceleration and pitch rate response at LANDING flight condition.

1006 IEEE TRANSACTIONS ON AUTOMATIC COKTROL, VOL. AC-31. NO. 1 1 . NOVEMBER 1986

TIME. ( $ 1

a -10

-12 -14

-- - --- ----- ---_______-_-- 0 2 0 40 60 80 100 120

'" C.G. LOCATION

186MAC ---- 50% MAC -

Fig. 35. Closed-loop speed response at LANDING flight condition

2 3

3 3 ,: 20 2 c 4 0 ; c b C - @

T I M E

Fig. 36. Normalized pitch rate response.


80

60

40

m 0 20 - z 9 0

-20 HIGH FREOUENC

do GAIN BOUNDARY

-60 .OD1 .o 1 1 1 10 100

80 c 1 I

60

40

m 0 20 - z 9 0

-20 HIGH FREOUEN

do GAIN BOUNDARY

-60 .OD1 .o 1 1 1 10 100

FREOUENCY, l r a d k l

.,,Ob I , , , I -360

,001 01 1 1 10 FREOUENCY. (radlrl

1 W

--- 18% WAC C.G. LOCATION.

- 50s MAC

Fig. 37. Elevator open-loop frequency response for LQG design.

Fig. 38. Stability margins and closed-loop roots-Case Study II

qualities equal to or better than those for current airplanes that have excellent inherent stability and control characteristics. Particularly important features were the task-tailored stick-force gradients. the lack of phugoid oscillations, and the invariant handling qualities across the c.g. range and flight envelope.

Case Study III: Control Law for a Highly Reliable Control System

This application involved the design of a control law for minimum safe flight. The purpose was to provide a simple, highly reliable. longitudinal control function in the event of loss of the more complex Eight control function described in Case Study II. The control law was obtained via direct synthesis of low-order feedforward and feedback control laws with constant gains. The airplane modeled is the same as that used in the previous case except that c.g. has been shifted further aft resulting in increased open-loop instabilities. When compared to an earlier control law derived using classical trial-and-error root locus techniques, this control law offered significant improvements in control performance and less sensitivity to variations in the airplane stability characteristics. The work was first reported in [32].

Problem Starement: Because of stringent reliability requirements. the design was constrained to: 1) one set each of redundant feedforward and feedback sensors, 2) simple feedforward and feedback compensation. 3) no integral control, 4) limited bandwidth due to the presence of unmodeled dynamics, and 5) fixed gains and filter parameters for the total flight envelope and all airplane configurations.

It was required that the control law should be implemented using a few highly reliable electronic components. Therefore, a simple control law structure consisting of first-order lead-lag filters in the feedforward and feedback paths, respectively, was selected. These filters would be implemented in a redundant set of hardware. Because of their high reliability. redundant sets of pitch rate and column force sensors were selected for the inputs to the feedback and feedforward loops. respectively.

Integral feedback control was not considered for the following reasons. First. integral control would be provided by the control law described in Case Study 11. Thus, including an integrator in the backup control law would lead to redundant and uncontrolla- ble integral modes if both control laws were operating simultane- ously. This problem could be avoided by having the backup control law as a standby mode. However, multichannel implementation of a control law with a pure integrator will require careful cross-channel synchronization. The required cross-channel signal paths could result in failures propagating from one channel to the other. thus compromising the required reliability.

In Case Study 11, nearly uniform command response characteristics were obtained for all flight conditions and airplane states. This was due to the high loop gain furnished by the feedback integrator. This cannot be achieved using the lead-lag compensator structure selected for this application. Therefore, the sensitivity or gain between the pilot's force input and the aircraft response will vary considerably. This variation will be a function of the

GAKGSAAS el at. : MODERN SYNTHESIS AND AIRCRAFT COXTROL 1007

open-loop airplane stability and control characteristics and the gains in the feedback and feedforward loops. The control law must ensure that the range of stick forces required to maneuver at various flight conditions and airplane states is manageable by the pilot.

Using pitch rate feedback it is only possible to significantly alter the short period mode characteristics of the airplane. For some flight conditions with the c.g. at the aft limit, the closed-loop airplane may exhibit instability in the form of a pure divergence. It occurs in the long-term speed and flight path response of the airplane and must be sufficiently slow to allow the pilot to maintain control.

Requirements and Objectives: The following are the main design requirements and objectives.

Satisfy minimum safe flying equalities requirement 121 and [26]. These are detailed specifications for allowable stability and output response Characteristics even with failures in the flight control system.

The maneuver stick-force gradients must be within 20 Ib!g and 120 Ib/g for flaps-up flight, and 30 Ib/g and 180 lbig for flaps down flight (landing).

The short period mode damping must be greater than 0.35. The transient pitch rate response must be within specified

The time-to-double amplitude of any unstable roots must

Gains and filter time constants must be constant. Meet f 6 dB gain margin and 2 45 deg phase margin within

Loop gain crossover frequency must be greater than:

envelopes (see Fig. 17).

exceed 12 s.

the control bandwidth.

Flaps-up flight: 2 rad/s Flaps-down flight: 1 rad/s

The high-frequency loop gain must be below - 10 dB at 10 rad/s with a minimum negative slope of - 40 dB!decade beyond 10 rad/s.

Design Method: The control law synthesis is based on the method described in [21]. It is a systematic approach to directly determine control law gains and filter parameters that will meet practical design constraints in terms of control law structure, performance objectives, and robustness requirements for plants represented by multiple linear models. The method 1211 is generally not known and therefore will be summarized here.

The procedure uses a nonlinear, constrained parameter optimization technique to calculate the control law parameters subjected to the differential constraints of multiple linear plant models. The latter are given by the following state-space descriptions:

x=A'x-+B'u+r'n (10)

x(t, j = x:, (1 1)

with outputs

J = C'X-+ D'u + Q'TZ (12)

for i = l . . . , Np.

The superscript i refers to the ith plant condition and Np is the total number of design conditions. The disturbances are random initial conditions with covariance:

and random white noise inputs n with covariance:

E[n(t+7)nT(f)]=N;8(7). (14)

The objective is to synthesize directly a control law that provides satisfactory stability, performance. and robustness at

several plant design conditions ( i = 1, . , A;). The control law has the general structure of a linear time-invariant state-space model expressed in the minimal realization form 1331:

i = A , z + BL.~15 (15)

u = C,.Z + Dry, (16)

where A,. is a block diagonal matrix. Any combination of independent parameters in the matrices A ( ,

BL., Cc, and D,. can be selected and used to minimize a quadratic cost function of the form:

.,\;

J(t ,)= 112 ~ ~ ' ~ , E [ ~ J ( t , ) Q ' l ; ( r , ) + u T ( r . ~ f ) R i ~ ~ ( t f ) ] . (17) I = 1

This weighted average cost function over I\> design conditions permits the designer to synthesize a linear time-invariant control law of the form expressed by (15) and (16). It satisfies performance and robustness requirements that represent a compromise between the various plant conditions.

The cost function is evaluated to a finite terminal time t-f. In contrast to using a steady-state cost function 1341. this approach does not require a stabilizing initial guess to start the optimization process. The steady-state solution (i.e.. when t , approaches infinity) is obtained by gradually increasing the value of rfuntil the final value of J(rr) settles to within a predetermined increment of its previous value. Convergence to a steady-state cost function automatically guarantees asymptotic stability when conditions of controllability and observability are satisfied.

The parameter optimization can also be conducted subject to additional linear and nonlinear constraints. Performing constrained optimization is useful since, in general. design requirements are not always easily expressed in the form of a quadratic cost function. With direct constraints. numerous iterative adjustments in the cost weighting matrices 0; and R' and the parameter Wpi to satisfy a given design requirement can be eliminated.

The constraints are of the form:

which represent linear or nonlinear inequality or equality constraints. and

which represents bounds on the covariances of certain outputs for given covariances of a set of disturbance inputs. Minimization of the cost function (17) subject to these constraints is performed numerically using a nonlinear programming technique based on a projected Lagrangian method [35].

Problem Formulation: The abwe design procedure was applied to the synthesis of both a feedback and feedforward control law. The flight conditions used are those shown in Fig. 11. At each condition there are two state models of the airplane representing the forward c.g. position (23 percent MAC) and the aft c.g. position (55 pcrcent MAC). respectively. The corresponding open-loop eigenvalues are shown in Fig. 39. At the worst condition the airplane is unstable with a time-to-double amplitude of less than 3 s ( h = 0.219).

The primary objective was to stabilize the short period mode and provide acceptable flying qualities [ 2 ] . The airplane model used in the synthesis was a short period approximation of the ful l airplane longitudinal dynamics. This approximation was obtained b j deleting the speed and pitch angle states from the models shown in Appendix C. However, closed-loop control analyses were performed with the full longitudinal models.

The control law structure is shown in Fig. 40. It is similar to that of a control law shown in Flg. 41 which had been synthesized earlier using the root locus technique, except that there are no gain

1008 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-31. NO. 11. NOVEMBER 1986

Fig. 39. Eigenvalues of the open-loop airplane-Case Study III.

I COLUMN FORCE Fc

changes as a function of flight condition and independent lead-lag filters have been added to the feedback and feedforward paths. respectively. Both control laws use pitch rate as the feedback signal and column force as the feedforward signal and accept commands from an additional control law such as the one described in Case Study 11. The state equations for the feedback and feedforward control laws are given by:

and

The feedback control law parameters are a, cI, and dl, and the feedforward control law parameters are c2 and dz. These five parameters define the gain and filter coefficients of the compensa- tors. A fixed 1 s time constant was specified for the feedforward path. z and z , are the feedback and feedforward compensator states, respectively. q is the pitch rate input, F, is the pilot's column force input. and 6, is the command to the elevator control servo.

The control law synthesis was performed in two steps. First the feedback parameters were calculated by solving a regulator problem. Given the airplane model with the feedback loop closed, the feedforward parameters were calculated by solving an explicit model-following problem.

The feedback control law parameters a, cl, and dl were

ri DRYDEN TUREULEKCE

Fig. 42. Synthesis model for direct reduced order design.

calculated to minimize the following cost function:

where nz is the airplane incremental normal acceleration response. The synthesis was performed using a model of the airplane at the VMIN flight condition (Fig. 11) with the c.g. at the most aft location. At this condition, the airplane exhibits the largest open- loop instability. It was found that satisfactory stability characteristics could be obtained for all flight conditions and c.g. locations without including the state models from the other flight conditions as differential constraints. As will be seen later, this was not possible in the feedforward design.

The cost function was evaluated for input disturbances consisting of vertical gust (Dryden spectrum [30]) w n ; control input noise 6,; and pitch rate sensor noise qn. The corresponding spectral densities are given in Fig. 42. Values were determined by performing a standard full-order LQG design with loop recovery using the short period mode model of the airplane. This design procedure was similar to that in Case Study 11. except that there was no integral control and only the pitch rate sensor was used in the estimator design. This control law met all design requirements and a reduced-order version of it furnished the initial values for the feedback parameters. Because of the computational ease of LQG synthesis, this approach was more efficient than attempting to determine the required spectral densities iteratively using the procedure in [2 11.

The cost function (22) does not include a penalty on the control 6,. In order to avoid infinite gains and bandwidth. the control law parameters were constrained as follows:

feedback filter pole: la1 s 10

direct pitch rate gain: Idll 5 10

gain on the filter state: IcI( 5 10.

To ensure that the variations in column force gradients were within 20 Ib/g and 120 lb/g for flaps-up flight conditions and 30 Ib/g and 180 Ib/g for flaps-down flight conditions. the static gain. in the feedback loop was constrained. Increasing static gain reduces variations in the column force gradient. The following lower bound was established based on the open-loop airpIane characteristics at all flight conditions and the specified requirements for maximum and minimum column force gradients:

static feedback loop gain: I c1 + d l I 2 3.15.

Direct constraint on the elevator rate covariance response was used to ensure sufficient loop gain attenuation at the higher frequencies. It was expressed as follows:

loop gain attenuation: ~ [ 8 : ( t , - ) ] 5 8: ma*. ( 2 3 )

GANGSxAS et ai.: MODERN SYNTHESIS AND AIRCRAFT CONTROL 1009

23% UAC I :w= 633 560 2 097~13 102 1 om:, 100 -3 03313 361

b P = 460 tw= 670 :sp= 480 743ijl 358 I r E + 00895(T2=76r' + OZWT7=32r! 0339*j0440 + 02921T2=23rl

555 YAC 0394 1292 384 2 550fll 580

123 419

187l f jZ 073 ?a= 670 tip- 538

104Ckll 629 7Olf11.036 I s = 850 ClP. 560

Fig. 43. Closed-loop eigenvalues for the modern design

I I * 020ii2-35rl - 0561T2=12r! 1 0282tl0960 - 0341T2=20r!

55% YAC

085 744 238

3 lEM15L12 3 033ii6 328

3 M&13 004

5 831 <rp= 785 :rp= 505 2 771

155 37 I

Fig. 4 4 . Closed-loop. eigenvalues for the classical design.

By lowering the upper bound $,,,. iteratively, the rolloff behavior at frequencies at and beyond 10 rad/s was adjusted until the requirements were satisfied

With the feedback control law defined, the feedforward control law was synthesized. The objective was to obtain good command response to column force inputs. The parameters c2 and d2 were determined from the minimization of the mean square error between the actual airplane incremental normal acceleration n, and the output of an ideal model z , ~ subjected to a step column force F, input. The cost function was of the form:

In order to meet command response requirements over the total flight envelope and all c.g. locations, the cost function had to be averaged over two flight conditions. These were VMIN and VFC/ MFC flight conditions (see Fig. 11). Both models were for the aft c.g., location (55 percent MAC). No constraints were imposed on the parameters c2 and dz, except they were scaled so that the lowest stick force gradient was 20 Ib/g. The feedforward controller design did not affect the stability of the closed-loop airplane. Block diagrams of the final feedforward and feedback control laws are shown in Fig. 40.

Results: Figs. 43 and 44 show the closed-loop eigenvalues at the various flight conditions for the direct reduced-order modem design and the classical design. respectively. Although they both meet the minimum requirement of 0.35 for the short period damping. the modern design exhibits better overall damping characteristics with considerably less sensitivity to changes in the flight condition. This is also true with regard to the aperiodic instability where the minimum time-to-double amplitude has been increased from 12-24 s. This improvement is due to higher low- frequency loop gain for the modem design when compared with the classical design.

Control loop stability margins are shown in Fig. 45. Both designs meet the minimum requirements. Fig. 46 demonstrates that the modern design meets the requirements for minimum crossover frequency and high frequency rolloff. The crossover frequency exceeds 2 rad!s for flaps-up CRUISE flight condition and 1 rad/s for flaps-down LANDING flight condition. The minimum gain attenuation requirements at high frequency are also

55% CAC

55% YAC

Claeleal design Modern design

tamn rmrgsn. Idegl Id81 ldegl Id6 I Phare mrgmn. Gan nargln. ?hare mlrglr

1 2 3 13 1

603 65 4

24 0 26 0

€ 4 0 56 0

1 4 4 17 3 57 0 30.0 77 0

51.0 25 0 61 0

10.9 11.6

56 8 61 6

73 0 24.0

70.0 59.0

27 8 23 0

109.3 102.6

39 0 39 0

84.0 65.0

Fig. 45. Minimum control loop stability margins-Case Study 111.

70

50 - 23% M A C

- 30 - ---- 55% M A C m --------- H I G H F R E Q U E N C Y 3 10 -

'k e, -1: - -30 -

LOW FREQUENCY -50 - G A I N B O U N D A R Y

-70 I I I

.o 1 1 1 10 100

- : 2 1 I , , ] -3w.l

.01 1 1 10 100 FREQUENCY, (radlrl

(a)

~

50 E-----

. - - 23% M A C --- 55% M A C -- - 30 2 10

- -- m

- ? .,x ---- H I G H F R E O U E N C Y

-30 - -50

-

-70

G A I N B O U N D A R Y LOW F R E Q U E N C Y

I I I

.o 1 1 1 10 100

. 3 q I , , I 360

.01 1 1 10 100

F R E Q U E N C Y , i r a d k )

(b) Fig. 4 6 , Modern design elevator open-loop frequency response. (a)

LANDING condition. (b) CRUISE condition.

met. These frequency responses. which are computed with the u, and 0 states deleted, are typical of those at the other flight conditions [32].

Fig. 47 shows the pitch rate responses at the LANDING and CRUISE flight conditions. Although both designs provide responses that fall within the envelopes, the modern design furnishes more desirable responses. The classical design exhibits considerable variations from flight condition to flight condition and a significant sensitivity to changes in the location of the c.g. In contrast. the modern design furnishes relatively invariant responses as the flight condition and c.g. location change. These responses are typical of those at the other flight conditions [32].

1010 IEEE TRA NSACTIONS ON AUTOMATIC CONTROL. VOL. AC-31. NO. 1 I . NOVEMBER 1986

C. G. LOCATIONS: - 23% MAC

55% MAC - _ 28

2 4

2.0

1.6

3 12 8

4

00 0 2 4 6 8 1 0 0 2 4 6 8 1 0

TIME. IS! TIME. 111

LANDINGCONDITION

0 2 4 6 8 1 0 0 2 4 6 8 1 0 TIUE. ( S I TIME ( $ 1

CRUISE CONDITION

Classical Design Modern Design

Fig. 47. Normalized pitch rate response-Case Study 111.

Co?JcLusroNs

Control law synthesis will play a key role in the design of new aircraft. It will be part of the iterative development of the aircraft configuration and propulsion system. The quality of the control laws will have a significant impact on overall aircraft performance and safety of flight. The best available synthesis and analysis techniques should therefore be used.

Recent developments in the practical application of modern control theory combined with the availability of excellent computer-based synthesis and analysis tools offer the potential for significant improvements in control law quality and reduction of the associated development cost. The three case studies are only modest examples of this. However, the results support the contention that significant benefits will accrue in the solution of control problems that have several control objectives and constraints. These are typical of most aircraft control problems that require careful blending of several outputs, inputs. or both to meet the objectives and satisfy the constraints.

In Case Study I. formulating the autopilot control law design as a linear quadratic regulator problem offered additional insight and understanding that were not available using the root locus technique. Specifically, the LQR synthesis allowed quick identifi- cation of the necessary feedback signals and corresponding gains required to achieve both good control performance and insensitivity to nonlinearities and parameter variations. Obtaining the same results using classical techniques would have required either more insight and experience on the part of the control en,' olneer or a more time-consuming and costly trial-and-error approach.

The key in solving this problem was to recognize the effects of simultaneous parameter variations in the rudder and aileron control loops. If multiloop sensitivity or stability margin analysis had been performed during the initial phase of control law development. the possibility of limit cycle oscillations would have been predicted prior to flight. This analysis could have been either in the form of introducing simultaneous parameter variations in the two control loops or in the form of the more formal singular value analysis [ 151.

Case Study I1 demonstrated the effectiveness of LQG synthesis when it is combined with classical frequency domain interpreta- tions of performance and sensitivity properties. The approach offered a direct way of incorporating requirements for command response, disturbance rejection, and insensitivity to plant model errors. Because each of these requirements were associated directly with parameters in the problem formulations, their implementation and the tradeoff between conflicting requirements were systematic and direct. The resulting control law maintained good performance and stability characteristics over a wide range of aircraft parameter variations. This made it possible to use a very simple gain scheduling scheme. In our opinion. using the classical synthesis techniques for this problem would have been more time consuming and led to a control law with more complex gain schedules. The latter would be required for more explicit compensation for known parameter variations.

The approach used in Case Study 111 offered a direct way of solving a very constrained control problem. There was a significant reduction in the number of required design iterations and an improvement in the quality of the control law when a comparison was made to earlier work performed using the standard root locus and frequency response techniques. The technique is computationally a more cumbersome procedure than classical as well as other modem synthesis techniques. However, the availability of efficient computer implementation of the design algorithms makes the approach cost effective for many applications.

The three case studies illustrate that modern control law synthesis techniques can be used to improve the quality and reduce development cost of control laws. However, as in the case of using classical techniques. success still requires good understanding of the control problem and the physical plant. Ignoring this fundamental premise is the cause of the so-called failures of modern techniques, exactly the same way it would be the cause of failure of any technique.

The new techniques are not radical departures from past practices. but are instead natural extensions and improvements to existing techniques. Competitive solutions of the control problems for future aircraft will require the application of many synthesis techniques including the new developments. For a given problem, the control engineer must be prepared to choose the most appropriate approach from a wide selection of alternatives.

A A c B B, C

APPENDIX A

NOMENCLATURE

Plant system matrix Controller system matrix Plant input matrix Controller input matrix Plant output matrix Controller output matrix Yawing moment coefficient due to aileron deflection Center of gravity Controller constraints Plant direct transmission matrix Controller direct transmission matrix Elevator input noise spectral density (deg)? s Expected value of [. ] Column force (Ib) Gravity constant (= 32.17 f t is ') Altitude rate ( f t i s ) Altitude rate sensor noise spectral density (ftk) '

Altitude rate sensor noise ( f t i s ) Yaw moment of inertia (slug ft') Linear combination of proportional and integral n,,, terms (g)

S

GANGSAAS ef al.: MODERN SYNTHESIS AND AIRCRAFT CONTROL 101 1

Cost function Ratio of calibrated airspeed to true airspeed (kn/ (ft/s)) Feedforward normal acceleration to column force coefficient (gllb) Integral nzu coefficient (s- I )

Ratio of short-term normal acceleration to long- term airspeed change &/knot) Proportional speed penalty on velocity (g/ft/s) Linear quadratic Gaussian Linear quadratic regulator Mean aerodynamic chord Disturbance input vector Disturbance covariance matrix Number of plant design conditions Lateral acceleration (g) Normal (vertical) acceleration (g) Normal acceleration sensor noise spectral density (g)? s Normal acceleration sensor noise (9) Blended normal acceleration and speed (g) Roll rate (deg/s) Criteria weighting matrix Pitch rate (deg/s) Dynamic pressure (Ib/ft2) Pitch rate sensor noise spectral density (deg/s)? s Pitch rate sensor noise (deg/s) Control weighting matrix Yaw rate (deg/s) Root mean square Laplace operator Wind shear rate input noise spectral density (ft/

Wind shear rate input noise (ft/s3) Time (s) Initial time (s) Finite terminal time for optimization Time-to-double amplitude Forward velocity (ft/s) Operating point for forward velocity (fils) Control input vector Incremental forward velocity (ft/s) Horizontal gust velocity (ft/s) Mean horizontal wind rate (ftls') Mean horizontal wind velocity (ft/s) Horizontal gust noise spectral density (fils)' s Horizontal gust noise input (ft/s) Longitudinal acceleration sensor noise spectral density (ft/s')' s Longitudinal acceleration sensor noise (ft/s') Lateral velocity (ft/s) Incremental lateral velocity (ft/s) Calibrated airspeed (kn) Flight condition defined in Fig. 11 Mean airspeed (ft/s) Flight condition defined in Fig. 11 Speed equation input noise spectral density (ft/

Speed equation input noise (ft/s') Reference airspeed (kn) True airspeed (fils) True airspeed sensor noise spectral density (ft/

True airspeed sensor noise (fils) Vertical velocity (ft/s) Incremental vertical body axis velocity (ft/s) Cost function weighting Vertical gust velocity (ft/s) Vertical gust variable Vertical gust noise spectral density (ft/s)' s

s3)2 s

s2) s

s ) l s

Vertical gust noise input (ft/s) Plant state vector Initial condition covariance matrix Plant initial condition Criterion variable (g) Dutch roll mode output Plant sensor output vector High-pass-filtered command signal (deg) Controller state Ideal model state Angle of attack (deg) Sideslip angle (deg) Plant disturbance input matrix Aileron position (deg) Aileron command (deg) Elevator position (deg) Elevator command (deg) Elevator noise input (deg) Flap position (deg) Servo noise input (deg) Servo position (deg) Damping ratio Pitch angle (deg) Eigenvalue rms value Roll angle (deg) Heading angle (deg) Track angle (deg) Complemented heading or track angle (deg) Disturbance direct transmission matrix Time shift (s) Frequency (rad/s) Estimated quantity Differentiation with respect to time

APPENDIX B

FLIGHT MECHANICS AND CONTROL DEFIWITIONS

This Appendix contains a brief description of the aeronautical terms used in this paper. For further information the reader is referred to sources such as [23] or [31]. Fig. 48 depicts a coordinate system for the equations of motion of an aircraft. A set of orthogonal axes Oxyz, where 0 is at the center of gravity, is fixed in the airplane and moves with it. U, V , and W are the velocity components of the center of gravity parallel to Ox, Oy, and 02, respectively, and p , q , and r are the angular velocities around the corresponding axes. p is the roll rate, q is the pitch rate, and r is the yaw rate of the aircraft.

In the equilibrium state Ox and Oz are in the vertical plane with Oz downwards. Ox may in general be at a nonzero angle with the horizontal plane. However, for simplicity this angle is assumed to be zero. Oy is in the horizontal plane. Fig 49 depicts the angular rotations that define the disturbed position of the axes. First OX,,^^^ is rotated about Ozo through the yaw angle $, next Oxlylzo is rotated about Oy, through the pitch angle 0, and finally Oxy,z2 is rotated about Ox through the roll angle 4. With these definitions of the yaw angle, pitch angle, and roll angle the roll rate, pitch rate, and yaw rate are defined as

p = d - $ sin 0

q = e cos & + $ cos B sin 6

r = - 6 sin 6-1-4 cos e cos 4 .

The general equations of the rigid body motions of the airplane are coupled and nonlinear with the state vector U, V, W , 0, 6, p , q , and r [23] or [31].

The equations used in this paper are a set of small perturbation equations linearized at the operating point CJ = U,, V = O . , W = O . , 0 = 0 . . 4 = O . . p = O . , q = O . , a n d r = O .

1012 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-31, NO. 11. NOVEMBER 1986

Y

RUDDER

ELEVATOR

Fig. 48. Airplane axes: Translational and rotational velocity components.

'\ -AILERON

RUDDER

ELEVATOR

Fig. 48. Airplane axes: Translational and rotational velocity components.

20

Fig. 49. Definition of the rotation angles.

The corresponding small elements of the perturbation state vector are ui, u , w, 8 , I$, p , q, and r. The linearization results in two decoupled sets of equations termed (1) the longitudinal equations of motion with the state vector u,, w, 8. and q; and ( 2 ) the lateral directional equations of motion with the state vector u. 4, P , and r.

The perturbed angle of attack cy is defined as

a = s i n - ' (z) =z 57.3 (deg).

The perturbed angle of sideslip p is defined as:

/3=sin-' (6) =v 57.3 (deg). UO

The incremental normal acceleration nz is defined as:

nz= Uo(q- &)/(57.3g0) (g).

The incremental lateral acceleration n, is defined as:

The altitude rate is defined as:

h = uo(e- 4 1 5 7 . 3 @IS).

The heading angle is equal to the yaw angle, $, and the track angle, $,, is defined as:

$rr = $ + P (deg).

The longitudinal motion of the airplane is characterized by the two second-order modes termed the short period and phugoid modes. For the type of aircraft considered in this paper the former is associated with states cy and q and range in frequency from 1 to 3 radls while the latter is associated with states ui and 8 and range in frequency from 0.05 rad/s to 0.2 radls.

The lateral directional motion of the airplane is characterized by the second-order dutch roll mode and the first-order roll and spiral modes. The motions involve perturbations in /3, F , p , and 4.

For the type of aircraft considered in this paper the frequencies are of the order of 1 rad/s for the dutch roll and roll modes and 0.01 rad/s for the spiral mode.

The location of the center of gravity (c.g.) is measured in terms of the distance along the mean aerodynamic chord (MAC) in units of percent MAC. The latter is a reference line located in the vertical plane of symmetry of the aircraft approximately parallel to the Ox axis. Its length and longitudinal position are a function of the shape, size, and position of the wing. The location of the c.g. has a strong influence on aircraft stability. As it moves aft, stability is reduced or instabilities are increased.

APPENDIX C MODELS

This Appendix contains model data for the aircraft of Case Studies I1 and 111. The equations are of the form

X=Ax+ Bu

Y=CX+DU

where x is n X 1 state vector, u is rn X 1 input vector, and y is p x 1 output vector. A , B, C, and D are n X n, n x m, p X n, and p X 177 matrices. respectively.

Disturbance Models

The Dryden turbulence state model is described in [30]. For Case Study 11, mean airspeed was defined as

v,,, = u - ~ , n w

where I/ is forward speed and un:,$ is horizontal mean wind speed. True airspeed is defined as

v, = u - U",,, - UR where u g is the zero mean horizontal random gust velocity. Mean airspeed V,,,, rather than true airspeed V,, was used in the regulated output for the control law synthesis in order to minimize control activity due to horizontal gust (u,) inputs.

State Model Data

The following are the data for the state models. The elements of the state. input and output vectors are defined in Appendix A.

case Study :: Flisht Ccndition: :3U:SE r ' t h i . 9 . at %I! HZC

4 Matrix: States ( ' J m , a, q, 9 , 5 , . ISerYO. u5. hg, Q', u,!

-.a1365 ,1760 .X017 -.5613 -.C3726 C. ,31365 -.0:311 0. -1. -.0!515 -.7520 1.X: ,3012' -.C631i C. ,31515 ,35536 0. .X:07 .37696 - .8:25 C . -3.399 0. -.30107 -,,X581 0.

0.

0. 0. 1. 0. 0.

0. 0. 0. c . 0. C. 0. 0. C . -2O.CO 10.72 C . C . 0.

0.

0. 0. c. c. 3. -5c.x 0. c . 0. 0. 0.

c . 0. 0. c. 0. 0. - .6347 0. 0. 0. 0. c. 0.

0.

c . n c . 3 . n.

0. 0. - .4447 ,0344 0 .

C. 0.

c . 0. 0.

0. 0. 0. 0.

S 0 . P

0. 0. 0. C. C. C.

c. e. 0.

.943:

0. 0. 3. 0.

0. 0. 0. 0. 0. 0. 0. 0. c. 0. 1.155 0.

c. .. -48.82 ?.

I. 3 . 3 . 3 . 3 . a. 3 . 0. 0. 0.

c . 0.

C. 0.

5c. C.

0. 0. 0. 0.

0. 0. 0.

C. C. C. 6. e. 0.

a.

0. 0. 0. 0. 3 . 0 . 0.

3 . 3 .

3 .

e. 0 . 3. 3 . 3. 3. 3. 3. 3. 0.

0. 0. 0. 0. 0. 0 . 0. 0. 0. 0.

0 . 0. 0. 0. C. 0. C. 0. 0. 0.

: Matrix: h t p u t i (ni. V T , ;i, h. ?!

.@C646 .3233 -.03346 G. -.1032 3 . -.WE85 -.C2358 3 . 0. i.C 0. e. 0. c . 3 . - : .0 3 . -.Cl361 .liaC, .C03i? -.56ic -.X726 3. .Cl365 -.C1311 3. 0.

3. 0.

1. -13.56 0. 1. e. :.

l3.58 0. 0. 3. c . 0. 0. 0 .

I. 3 . 0. _. 3. 0.

I qztrix: :-pLts (aec. L;, rn. I - . '8". I?:~. J : , . in, 1:". :r. /II: 3 . 0. 3 . 3 . 0 . 3 . 0 . 0 . 1 . 0 . 0 . 3. 3. 3 . 3 . , : . 0 . 1 . 0 . c . 0 . 3 . 3. 3. 0. o . c . 0 . 0 . 1 . c . 3 . 3 . 3. 3. 0. o . c . 3 . 3. 0 . c . 3 . 1 . 3. 3. 0 . 0 . , : . 3 . 0. 0. c.. 1 . 1 .

GANGSAAS et al.: MODERN SYNTHESIS AND AIRCRAFT CONTROL 1013

Case Study i l i

1) Flight Condition: VFCIHFC uith C.9. at 55% WC

A Matrix: States (ui. a, 4. 9 , 6?, dseryo. ug. “9, wg?)

-.00702 .06339 .00518 -.55566 -.06112 0. ,00712 -.00566 0. -.OK54 -.38892 1.0057 ,00591 -.04632 0. .01554 .04C18 0.

.00061 .35210 -.47381 -.OOOOO 1.7862 0. -.00061 -.a3638 C. 0. 0. 0. 0. 0. 0.

0. 0. 0. 0. 0. 0.

0. 1.

0. 0. 0. 0.

0.

0. 0.

0. 0. 0.

-20. 0.

0.

0. 0.

0.

8 Matrix: Inputs (6ec, un, wn, 6,. qn)

20. 0.

-30. 0. 0. 0.

0. 0. 0. 0. 0. 0. 0. 0. 0. -.5545& 0. 0. 0. -.55455 .On555 0. -.XI555 -.55454

C Matrix: Output (nZ. q)

.W5W .11679 -.On172 .C€OOO -.31413 0. -.0050C -.01207 0. 0. 0. !.O 0. 0. 0. 0. 0. 0.

0 Hatrix: Inputs (dec, u,, w,, 6,. qn)

0. 0. 0. 0. 0. 0. 0. 0. 0. 1.

Case Study I:!

2) Flight Condition: VHIY with c.g. at 55% HAC

4 Matrix: States (ui, 2. q. 3, I , , iserlO. u9. ws, ugl)

-.06254 .0188@ .OOOOO -.56141 -.02751 0. .06254 -.IN123 0. .01089 -.99290 .99795 .ON97 -.07057 0. -.01089 .06449 0.

0. 0. 1. 0. 0. 0. 0. 0. 0. -20.

0. 0. 0. 0. 20. 0. 0. 0.

0. 0. 0. 0. a. -30. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. a.

0. -.E8206 0. 0.

0. 0. I . 0. 0.

0. 0.

0. 0. -.8@206 .00882 0. 0. -.00882 -.88206

8 Yatrix: lnouts ( 6 e c , u,, an, 5 , . qn)

.a7743 1.6754 -1.3111 -.no030 -4.2503 0. -.on43 -.loem 0.

0. 0. 0. 0. 0 . 0. 0.

0. 0.

0. 0. 0.

30. 0. 0. 0. 0.

3. 0. !.3282 0. 0. 0. 1.62671 0. 0. -68.75283

C Hatr’x: Output (nz, 4)

0. 0. 1. 0. 0. -.OX19 .17604 .OCC9@ -.CUI31 .03378 0. .00519 -.Oh186 0.

0. 0. 0. 0.

0 Yatrir: ilpbts ( s e e c . u,, v,, 6,. qn)

c. 0. 0. 0. 0. 0. 0. 0. 0. 1.

r11

REFERENCES

W. Bollay. The Fourteenth Wright Brothers Lecture-“Aerodynamic stability and automatic control,” J. Aeronautical Sci., vol. 18, Sept. 1951. Military Specification: “Flying qualities of piloted airplanes,” MIL-F- 8785C, Nov. 1980. W. M. Bode, Network Analysis and Feedback Amplifer Design. Princeton, NJ: Van Nostrand, 1945. W. R. Evans, “Control system synthesis by root locus method,” Trans. AIAA, vol. 69, pp. 66-69, 1950. I . M . Horowitz, Synthesis of Feedback System. New York: Academic, 1963. H. H. Rosenbrock, Computer-Aided Control System Desim. New York: Academic, 1974. A. G. J. MacFarlane and B. Kouvaritakis, “A design technique for linear multivariable feedback systems,” Int. J. Contr., vol. 25. pp. 837-879. 1977. M. Athans and P. L. Falb, Optimal Control. New York: McGraw- Hill. 1966. A. E. Bryson and Y. C. Ho. Applied Optical Control. Waltham,

B. D. 0. Anderson and J. B. Moore. Linear Optimal Control. MA: Blaisdell: 1969.

Englewood Cliffs. NJ: Prentice-Hall, 1971. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems.

B. D. 0. Anderson and J. B. Moore. Optimal Filtering. Englewood New York: Wiley, 1972.

Cliffs, NJ: Prentice-Hall. 1979.

~~~ ~ ~~

1221

r231

1241

Special Issue on the LQG Problem, IEEE Trans. Automat. Contr., Dec. 1971. W. A. Wolovich, Linear Multivariable System. New York: Springer-Verlq, 1974. J. C. Doyle, “Multivariable design techniques based on singular value generalizations of classical control,” AGARD GRF’ Lecture Series 117, Oct. 1981. J. C. Doyle and G. Stein, “Multivariable feedback design: Concepts for a classical/modern synthesis,” IEEE Trans. Automat. Contr., Feb. 1981. G. Stein and M. Athans, “The LQG/LTR procedure for multivariable feedback control design,” Mass. Inst. Technol., Cambridge, MA, Rep. LIDS-P-1384, May 1984. M. G. Safonov at$ M. Athans, “Gain and phase margins of multi-loop

R. E. Kalman, “When is a linear system optimal?” Trans. ASME Ser. LQG regulators, IEEE Trans. Automat. Contr., Apr. 1977.

D; J. Basic Eng., vol. 86, pp. 5 1-60, 1964. J. C. Doyle and G. Stein, “Robustness with observers,” IEEE Trans. Automat. Contr., vol. AC-24, no. 4, pp. 607-611, Aug. 1979. U. Ly, ”A design algorithm for robust low order controllers,” Dep. Aeronaut. and Astronaut., Stanford Univ., Stanford, CA, Rep. 536, Nov. 1982. K. R. Bruce and D. Gangsaas, “Improvement of the 767 lateral autopilot using optimal control design techniques,” presented at the AIAA Guidance and Control Conf.. Seattle, WA, Aug. 1984. A. W . Babister, Aircraft Stability and Control. New York:

J. D. Blight, D. Gangsaas, and T. M. Richardson, “Control law Pergamon, 1961.

synthesis for an airplane with relaxed static stability.” AIAA J. Guidance, Contr., and Dynam. (Special Issue on Handling Quali- ties), Sept.-Oct. 1986. NASA CR-159249, ”Integrated application of active controls (IAAC) technology to an advanced subsonic transport project-Initial ACT

July 1980. configuration design study,” Boeing Commercial Airplane Company,

FAR 25, Federal Aviation Administration, 1978. D. Gangsaas, U. Ly, and D. C. Norman, “Practical gust load alleviation and flutter suppression control laws based on LQG methodology,” presented at the AIAA 19th Aerospace Sci. Meet., St. Louis, MO, Jan. 1981, paper AM-81-0021. N. A. Lehtomaki, N. R. Sandell, Jr., and M. Athans, “Robustness

design,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 75-92, results in linear quadratic Gaussian based multivariable control

Feb. 1981. N. K. Gupta, “Frequency-shaped cost function&: Extension of linear- quadratic-Gaussian design methods,” AIAA J. Guidance Contr., vol. 3, pp. 529-535, Nov.-Dec. 1980. N. M. Barr, D. Gangsaas, and D. R. Schaeffer, “Wind models for flight simulator certification of landing and approach guidance and control systems,” Dec. 1974, paper FAA-RD-74-206. J. Roskam, “Airplane dynamics and automatic fight controls,”

U. Ly, “Optimal low order flight critical pitch augmentation control Roskam Aviation and Engineering Corporation, 1979.

law for a transport airplane,“ AIAA Guidance Contr. Conf., Seattle, WA, Aug. 1984. T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980.

[34] E. J . Davison and I. Ferguson, “The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods,” IEEE Trans. Automat. Contr., vol. 26, no. 1, pp. 93-1 10, 1981.

[35] P. E. Gill, W. Murray, and M. Wright, Practical Optimization. New York: Academic. 1981.

1014 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-31, NO. 11, NOVEMBER 1986

Systems Research Depad for the 7J7 program.

Kevin R, Bruce received the B.S. degree in mechanical engineering in 1973 from Liverpool Polytechnic, Liverpool, England, and the M.S. degree in control engineering in 1978 from Hatfield Polytechnic, England.

He was employed by the British Aerospace Guided Weapons Division from 1973 to 1979, where he worked in the Theoretical Studies Department. Since 1979 he has been employed by the Boeing Commercial Airplane Company, Seattle, WA, where until 1985, he worked in the Flight

:ment. Currently he is working on system integration

James D. Blight received the B.S.E. degree in electrical engineering and the Ph.D. degree in computer, information, and control engineering from the University of Michigan, Ann Arbor.

He has worked with TRW Systems, Redondo Beach, CA, on spacecraft control and Tektronix, Inc., Beaverton, OR, on electronic circuit design. Since 1978, he has been with The Boeing Company, Seattle, WA, where he has worked on applications of modern control design to flight control systems for commercial and military aircraft. He has also

served on the adjunct faculties of Oregon State University and the University of Portland.

Dr. Blight is a member of Tau Beta Pi, Eta Kappa Nu, and Phi Kappa Phi.

ieee transactions on application of modern synthesis to aircraft control: three...

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